THE SQUARING OF THE CIRCLE. 



119 



solving the problem. This question is to be answered in the nega- 

 tive. For in geometry there can easily exist pairs of lines of which 

 the one can be readily constructed from the other, notwithstanding 

 the fact that no two whole numbers can be found to represent the 

 ratio of the two lines. The side and the diagonal of a square, for 

 instance, are so constituted. It is true the ratio of the latter two 

 magnitudes is nearly that of 5 to 7. But this proportion is not 

 exact, and there are in fact no two numbers that represent the ratio 

 exactly. Nevertheless, either of these two lines can be readily con- 

 structed from the other by employing only straight edge and com- 

 passes. This might be the case, too, with the rectification of the 

 circle ; and consequently from the impossibility of representing 7t 

 by the ratio between two whole numbers the impossibility of the 

 problem of rectification is not inferable. 



The quadrature of the circle stands and falls with the problem 

 of rectification. This rests upon the truth above mentioned, that 

 a circle is equal in area to a right-angled triangle, in which one 

 side is equal to the radius of the circle and the other to the circum- 

 ference. Supposing, accordingly, that the circumference of the circle 

 had been rectified, then we could construct this triangle. But every 

 triangle, as we know from plane geometry, can, with the help ol 

 straight edge and compasses be converted into a square exactly 

 equal to it in area. So that, supposing the rectification of the cir- 

 cumference of a circle to have been successfully effected, a square 

 could be constructed that would be exactly equal in area to the 

 circle. 



The dependence upon one another of the three problems of the 

 computation of the number n, the quadrature of the circle, and its 

 rectification, thus obliges us, in dealing with the history of the 

 quadrature, to regard investigations with respect to the value of TC 

 and attempts to rectify the circle as of equal importance, and to 

 consider them accordingly. 



We have used repeatedly in the course of this discussion the 

 expression " to construct with straight edge and compasses." It 

 will be necessary to explain what is meant by the specification of 

 these two instruments. When to a requirement in geometry to 



